Linear Algebra Answers

Let λ ∈ R be an eigenvalue of an orthogonal matrix A. Show that λ = ±1.
(Hint: consider the norm of Av, where v is an eigenvector of A associated with the
eigenvalue λ.)

Also, find diagonal orthogonal matrices B, C such that 1 is an eigenvalue of B
and −1 is an eigenvalue of C.

Show that f(1, 1, 1), (1, – 1, 0), (1, 1, 0)1 is a
basis of R3 . Find a dual basis to this basis.

Let W {(x, y, z) R3: x + y + z = 0}. Check
if W is a subspace of R3 . Find a non-zero
subspace U of R3 so that W(intersection)U = (0).

Reduce the quadratic form 8x²— 4xy + 5y² to
its normal canonical form.

Consider linear operator T:C^3-->C^3 def by T(z1,z2,z3)=(z1+iz2, iz1-2z2, -iz2+z3). Check T* and check if T is self adjoined. check if T is unitary

Check whether following system of equations has a solution.
3x+2y+6z+4w=4
x+2y+2z+w=5
x+z+3w=3

Define T:R^3-->R^3 by T(x,y,z)=(-x,x-y,3x+2y+z). Check if T satisfies the polynomial (x-1)(x+1)^2.

Let T:R^3--->R^4 be defined by T(x1,x2,x3)=(x1+x2, x2+x3, x1-x3, 2x1+x2-x3). Check T is operator. Find kernel and range of T. Find dimensions of kernel.

Let U,V be subspaces of R^n.
Show that U∩V={0} if and only if S∪T is a linearly independent set of vectors for every linearly independent set S={u1,u2,...uk} ⊆ U and every linearly independent set T = {v1,v2,...vl} ⊆ V

Let V be the solution space of the following homogeneous linear system: x1 − x2 − 2x3 + 2x4 − 3x5 = 0 x1 − x2 − x3 + x4 − 2x5 = 0. (a) (2 points) Find a basis S of V and write down the dimension of V.
(b) (3 points) Finda subspace W of R5 suchthat W contains V anddim(W) = 4. Justify your answer.